Compatible extension of the (G′/G)-expansion approach for equations with conformable derivative

In this study, the compatible extensions of the (G′/G)-expansion approach and the generalized (G′/G)-expansion scheme are proposed to generate scores of radical closed-form solutions of nonlinear fractional evolution equations. The originality and improvements of the extensions are confirmed by their application to the fractional space-time paired Burgers equations. The application of the proposed extensions highlights their effectiveness by providing dissimilar solutions for assorted physical forms in nonlinear science. In order to explain some of the wave solutions geometrically, we represent them as two- and three-dimensional graphs. The results demonstrate that the techniques presented in this study are effective and straightforward ways to address a variety of equations in mathematical physics with conformable derivative.


Introduction
In contemporary years, nonlinear fractional evolution equations (NFEEs) have appealed the attention of the researchers for their importance in describing the internal processes of real-world incidents. NFEEs appear in various technical and industrial arenas, for instance, plasma physic, electricity, fluid mechanics, modelling of earthquake, quantum physics, chemical kinematics, water wave mechanics, etc. Traveling-wave solutions of NFEEs are crucial for understanding the underlying workings of complicated physical processes. Thus, several methods for extracting explicit wave solutions have been developed by researchers. For example, the inverse scattering approach [1], the Hirota's bilinear transformation [2], the Painleve expansion in truncated type [3], the Jacobi elliptic function approach [4,5], the Weierstrass elliptic function scheme [6], the fractional Riccati equation approach of the Bäcklund transformation [7], exponential-function technique [8,9], tanh-function approach [10,11], the systematic approach [12], and others [13].
Diverse academics have conducted further research to demonstrate the validity and efficacy of the (G ′ /G)-expansion approach and to widen its scope of applicability; such as, Zhang et al. [23] suggested the improved (G ′ /G)-expansion scheme. In the elementary method, the solution was offered in the shape u(ξ) = ∑ n j=0 a j (G ′ /G) j , where a n ∕ = 0, but in Ref. [23] Zhang et al. used u(ξ) = ∑ n j=− n a j (G ′ /G) j as a solution. Many researchers studied various NFEEs to determine exact solutions expending the improved (G ′ /G)-expansion method which can be found in Refs. [24][25][26][27]. Guo and Zhou [28] suggested the extended (G ′ /G)-expansion } and investigated exact solutions to the Whitham-Broer-Kaup-like equation and coupled Hirota-Satsuma KdV equations. Later, different researchers applied this extended approach to construct different classes of traveling wave solutions for some NFEEs [29,30]. In order to obtain exact results of the KdV equation, the stress equation for micro-structured objects and the ZKBBM equation, Akbar et al. [31] suggested a comprehensive and enhanced (G ′ /G)-expansion methodology, wherein the solution is offered in the formula u(ξ) = ∑ n j=− n c− j (d+(G ′ /G)) j , where together c − n and c n should not be zero consecutively. Naher and Abdullah [32] proposed the generalized (G ′ /G)-expansion process in which the result was offered by the formula u(ξ) = ∑ n j=0 a j (d + G Since each nonlinear fractional equation has its specific meaningful rich structure, there is still a lot of work to be done before the original (G ′ /G)-expansion approach can be fully established. Therefore, in this investigation, we suggest a development of the (G ′ /G)-expansion approach and the generalized (G ′ /G)-expansion approach in the case of the conformable fractional derivative suitable for searching NFEEs. To validate the novelty and suitability of the proposed extensions, the space-time fractional CB equations are exposed and assorted descriptive traveling wave solutions are uncovered.

Algorithm of the extensions
Let us assign a general NFEE of the shape: where D α t u, D β x u, D γ y u and D δ z u indicate conformable derivatives of the wave function u relating to x, y, z and t.
Step 1. Combine the temporal variable t and spatial variables x, y and z by the wave coordinate ξ as: where K, N and M are non-zero constraints and C is the wave transmission rate. Eq. (1) is transformed by the fractional wave conversion Eq. (2) into a nonlinear equation of classical derivative for u = u(ξ): where F is a nonlinear function of u(ξ) and its total derivatives.
Step 2. Eq. (3) will be integrated based on the possibilities.

Extension of the (G ′ /G)-expansion method
Step 3. Assume that using the presented approach, the estimation of Eq. (3) is given ensuing: where any one of a n or b n might be vanished, a j (j = 0, 1, 2, …, n), b j (j = 1, 2, 3, …, n) and d are subjective parameters to be calculated afterward, and φ(ξ) is specified below: where G = G(ξ) fulfills the subsequent second-order equation: where λ and μ are constraints.
Step 4. The assessment of n in Eq. (4) can be obtained by applying the homogeneous balancing principle between the maximal-order derivative and nonlinear extents in equation (3).
It is obtained following three solutions of equation (5) expending the generalized estimations of equation (6): where A 1 and A 2 are subjective constants.
Step 6. Assume that the assessment of the coefficients a j (j = 0, 1, 2, …, n), b j (j = 1, 2, 3, …, n), d, K, N, M and C can be gained by resolving the equations ascertained in Step 5. The constant measures, along with the generic estimations of (6), are then substituted into Eq. (4), yielding new general type and expedient solutions of the NFEE (1).

Extension of the generalized (G ′ /G)-expansion method
Step 7. As per this technique the solution of Eq. (3) is presented in the form: where either a n or b n might be zero, a j (j = 0,1,2,…,n), b j (j = 1, 2, 3, …, n) and d are parameters to be computed well along, and φ(ξ) is in the form: where G = G(ξ) satiates the ensuing second-order nonlinear equation: where A, B, E, D are constraints and prime denotes the derivative relating to ξ.
Step 8. It is ascertained that the integer n can be found from Step 4. It can be obtained the resulting five solutions of Eq. (11) by means of the generalized solutions of Eq. (12).
where A 1 and A 2 are integral constants.
Step 10. Assume that the assessment of parameters a j (j = 0, 1, 2, …, n), b j (j = 1, 2, 3, …, n), d, K, N, M and C can be obtained by unravelling the algebraic equations found in Step 9. Afterward, substituting the constant values together with the general solutions of Eq. (12) into (10) yields more broad-spectrum solutions and fresh solutions of the NFEE (1).

Application of the suggested extensions
In this segment, we put on the suggested extensions to examine the coupled time-space fractional CB equations. The time-space fractional CB equation [17,36] is: where 0 < α < 1 and the system parameters, such as Brownian diffusivity and the Stokes speed of the particle owing to gravity, affect parameters p and q. Now using the definition for fractional wave variable v(x,t) = v(ξ), u(x, t) = u(ξ) and ξ = K x α α + C t α α into Eqs. (19) and (20) yields:

The extension of the generalized (G ′ /G)-expansion method
Likewise, by means homogeneous balance between uu ′ and u ′′ in Eq. (20); vv ′ and v ′′ in Eq. (21), yield n 1 = n 2 = 1. Therefore, the solution's shapes are similar to those of (22) and (23), and for simplicity, they are not repeated here. Therefore, stand in for solutions (22) and (23) along with (11) and (12) into Eqs. (20) and (21), the left-hand sides are adapted into polynomial in (d + 1/φ) n , (n = 0, 1, 2, …) and (d + 1/φ) − n , (n = 1, 2,…). Collecting the coefficients of like power of the polynomial to zero, yield two sets of algebraic equations (which are untaken here for easiness) for a 0 , a 1 , b 1 , c 0 , c 1 , d 1 , d, K and C. We resolve the algebraic equations with the help of Maple software package, we obtain four different sets of solutions as follows: Set 1: where K, a 0 , A, B, d and E are random constants.
we obtain hyperbolic function solutions as: we obtain trigonometric function solutions as: where a 0 , K, d, E, D, A, and B are random parameters.
If B ∕ = 0, Ω = B 2 + 4E(A − D) > 0, we obtain hyperbolic function solutions as: we obtain trigonometric function solutions as: we obtain rational form solutions as: .
we obtain hyperbolic function solutions as: ) .
If B ∕ = 0, Ω = B 2 + 4E(A − D) > 0, we obtain hyperbolic function solutions as: we obtain trigonometric function solutions as: we obtain rational form solutions as: we obtain hyperbolic function solutions as: If B = 0, Δ = E(A − D) < 0, we obtain trigonometric function solutions as:

The numerical simulations
The wave assessments to the time-space fractional CB equations are numerically simulated in this section. In Figs Fig. 6 within the domain 0.1 ≤ t ≤ 60 and − 60 ≤ x ≤ 60.

Validation and advantages
The validity and advantages of proposed extensions over the original (G ′ /G)-expansion scheme and the generalized (G ′ /G)-expansion technique are deliberated in the underneath.

Conclusion
To establish abundant novel and wide-spectral general soliton solutions of NFEE solutions, extensions of the (G ′ /G)-expansion  method and the generalized (G ′ /G)-expansion method are presented and implemented in this article. Many new exact general solutions have resulted from these extensions, which provide distinct physical structures with free parameters. In order to illustrate the advantages and validity of the algorithms, we exploit these extensions to the space-time fractional CB equations. To retrieve the wellknown and general solutions, we portray the three-and two-dimensional graphs. The structures are capable of explaining the physical phenomena modulated by the space-time fractional CB equations. The obtained results are compared with those accessible in the literature, and it is revealed that these extensions are encouraging mathematical tools compared to existing ones. The results show that the extensions proposed in this article are straightforward, efficient, and applicable to a wide-range of fractional differential equations in mathematical physics.

Author contribution statement
Altaf A. Al-Shawba: Conceived and designed the experiments; Contributed reagents, materials, analysis tools, or data; Wrote the paper.
Farah A. Abdullah: Performed the experiments; Analyzed and interpreted the data; Contributed reagents, materials, analysis tools, or data.
Amirah Azmi, M. Ali Akbar: Analyzed and interpreted the data; Contributed reagents, materials, analysis tools, or data. Kottakkaran Sooppy Nisar: Contributed reagents, materials, analysis tools, or data; wrote the paper.

Data availability statement
No data was used for the research described in the article.

Declaration of Competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.